1951 AHSME Problems

Problem 1

The percent that $M$ is greater than $N$ is:

$\mathrm{(A) \ } \frac {100(M - N)}{M} \qquad \mathrm{(B) \ } \frac {100(M - N)}{N} \qquad \mathrm{(C) \ } \frac {M - N}{N} \qquad \mathrm{(D) \ } \frac {M - N}{M} \qquad \mathrm{(E) \ } \frac {100(M + N)}{N}$

Solution

Problem 2

A rectangular field is half as wide as it is long and is completely enclosed by $x$ yards of fencing. The area in terms of $x$ is:

$(\mathrm{A})\ \frac{x^2}2 \qquad (\mathrm{B})\ 2x^2 \qquad (\mathrm{C})\ \frac{2x^2}9 \qquad (\mathrm{D})\ \frac{x^2}{18} \qquad (\mathrm{E})\ \frac{x^2}{72}$

Solution

Problem 3

If the length of a diagonal of a square is $a + b$ , then the area of the square is:

$\mathrm{(A) \ (a+b)^2 } \qquad \mathrm{(B) \ \frac{1}{2}(a+b)^2 } \qquad \mathrm{(C) \ a^2+b^2 } \qquad \mathrm{(D) \ \frac {1}{2}(a^2+b^2) } \qquad \mathrm{(E) \ \text{none of these} }$

Solution

Problem 4

A barn with a flat roof is rectangular in shape, $10$ yd. wide, $13$ yd. long and $5$ yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. The total number of sq. yd. to be painted is:

$\mathrm{(A) \ } 360 \qquad \mathrm{(B) \ } 460 \qquad \mathrm{(C) \ } 490 \qquad \mathrm{(D) \ } 590 \qquad \mathrm{(E) \ } 720$

Solution

Problem 5

Mr. A owns a home worth $10,000$ dollars. He sells it to Mr. B at a $10 \%$ profit based on the worth of the house. Mr. B sells the house back to Mr. A at a $10 \%$ loss. Then:

$\textrm{(A)}\ \text{A comes out even} \qquad\textrm{(B)}\ \text{A makes 1100 on the deal} \qquad\textrm{(C)}\ \text{A makes 1000 on the deal}$ $\textrm{(D)}\ \text{A loses 900 on the deal} \qquad\textrm{(E)}\ \text{A loses 1000 on the deal}$

Solution

Problem 6

The bottom, side, and front areas of a rectangular box are known. The product of these areas is equal to:

$\textrm{(A)}\ \text{the volume of the box} \qquad\textrm{(B)}\ \text{the square root of the volume} \qquad\textrm{(C)}\ \text{twice the volume}$ $\textrm{(D)}\ \text{the square of the volume} \qquad\textrm{(E)}\ \text{the cube of the volume}$

Solution

Problem 7

An error of $.02"$ is made in the measurement of a line $10"$ long, while an error of only $.2"$ is made in a measurement of a line $100"$ long. In comparison with the relative error of the first measurement, the relative error of the second measurement is:

$\mathrm{(A) \ } \text{greater by }.18 \qquad\mathrm{(B) \ } \text{the same} \qquad \mathrm{(C) \ } \text{less} \qquad\mathrm{(D) \ } 10\text{ times as great} \qquad\mathrm{(E) \ } \text{correctly described by both}$

Solution

Problem 8

The price of an article is cut $10 \%.$ To restore it to its former value, the new price must be increased by:

$\mathrm{(A) \ } 10 \% \qquad\mathrm{(B) \ } 9 \% \qquad \mathrm{(C) \ } 11\frac{1}{9} \% \qquad\mathrm{(D) \ } 11 \% \qquad\mathrm{(E) \ } \text{none of these answers}$

Solution

Problem 9

An equilateral triangle is drawn with a side length of $a.$ A new equilateral triangle is formed by joining the midpoints of the sides of the first one. then a third equilateral triangle is formed by joining the midpoints of the sides of the second; and so on forever. the limit of the sum of the perimeters of all the triangles thus drawn is:

$\mathrm{(A) \ } \text{Infinite} \qquad\mathrm{(B) \ } 5\frac{1}{4}a \qquad \mathrm{(C) \ } 2a \qquad\mathrm{(D) \ } 6a \qquad\mathrm{(E) \ } 4\frac{1}{2}a$

Solution

Problem 10

Of the following statements, the one that is incorrect is:

$\textrm{(A)}\ \text{Doubling the base of a given rectangle doubles the area.}$ $\textrm{(B)}\ \text{Doubling the altitude of a triangle doubles the area.}$ $\textrm{(C)}\ \text{Doubling the radius of a given circle doubles the area.}$ $\textrm{(D)}\ \text{Doubling the divisor of a fraction and dividing its numerator by 2 changes the quotient.}$ $\textrm{(E)}\ \text{Doubling a given quantity may make it less than it originally was.}$

Solution

Problem 11

The limit of the sum of an infinite number of terms in a geometric progression is $\frac {a}{1 \minus{} r}$ (Error compiling LaTeX. Unknown error_msg) where $a$ denotes the first term and $\minus{} 1 < r < 1$ (Error compiling LaTeX. Unknown error_msg) denotes the common ratio. The limit of the sum of their squares is:

$\textrm{(A)}\ \frac {a^2}{(1 \minus{} r)^2} \qquad\textrm{(B)}\ \frac {a^2}{1 \plus{} r^2} \qquad\textrm{(C)}\ \frac {a^2}{1 \minus{} r^2} \qquad\textrm{(D)}\ \frac {4a^2}{1 \plus{} r^2} \qquad\textrm{(E)}\ \text{none of these}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Problem 12

At $2: 15$ o'clock, the hour and minute hands of a clock form an angle of:

$\textrm{(A)}\ 30^{\circ} \qquad\textrm{(B)}\ 5^{\circ} \qquad\textrm{(C)}\ 22\frac {1}{2}^{\circ} \qquad\textrm{(D)}\ 7\frac {1}{2} ^{\circ} \qquad\textrm{(E)}\ 28^{\circ}$

Solution

Problem 13

$A$ can do a piece of work in $9$ days. $B$ is $50\%$ more efficient than $A$ . The number of days it takes $B$ to do the same piece of work is:

$\textrm{(A)}\ 13\frac {1}{2} \qquad\textrm{(B)}\ 4\frac {1}{2} \qquad\textrm{(C)}\ 6 \qquad\textrm{(D)}\ 3 \qquad\textrm{(E)}\ \text{none of these answers}$

Solution

Problem 14

In connection with proof in geometry, indicate which one of the following statements is incorrect:

$\textrm{(A)}\ \text{Some statements are accepted without being proved.}$ $\textrm{(B)}\ \text{In some cases there is more than one correct order in proving certain propositions.}$ $\textrm{(C)}\ \text{Every term used in a proof must have been defined previously.}$ $\textrm{(D)}\ \text{It is not possible to arrive by correct reasoning at a true conclusion if, in the given, there is an untrue proposition.}$ $\textrm{(E)}\ \text{Indirect proof can be used whenever there are two or more contrary propositions.}$

Solution

Problem 15

The largest number by which the expression $n^3-n$ is divisible for all possible integral values of $n$ , is:

$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

Solution

Problem 16

If in applying the quadratic formula to a quadratic equation

$f(x) \equiv ax^2 + bx + c = 0,$

it happens that $c = \frac{b^2}{4a}$ , then the graph of $y = f(x)$ will certainly:

$\textbf{(A)}\ \text{have a maximum}\qquad\textbf{(B)}\ \text{have a minimum}\qquad\textbf{(C)}\ \text{be tangent to the x-axis}\\ \qquad\textbf{(D)}\ \text{be tangent to the y-axis}\qquad\textbf{(E)}\ \text{lie in one quadrant only}$

Solution

Problem 17

Indicate in which one of the following equations $y$ is neither directly nor inversely proportional to $x$ :

$\textbf{(A)}\ x+y = 0\qquad\textbf{(B)}\ 3xy = 10\qquad\textbf{(C)}\ x = 5y\qquad\textbf{(D)}\ 3x+y = 10$

Solution

Problem 18

The expression $21x^2+ax+21$ is to be factored into two linear prime binomial factors with integer coefficients. This can be one if $a$ is:

$\textbf{(A)}\ \text{any odd number}\qquad\textbf{(B)}\ \text{some odd number}\qquad\textbf{(C)}\ \text{any even number}\qquad\textbf{(D)}\ \text{some even number}\qquad\textbf{(E)}\ \text{zero}$

Solution

Problem 19

A six place number is formed by repeating a three place number; for example, $256256$ or $678678$ , etc. Any number of this form is always exactly divisible by:

$\textbf{(A)}\ 7\text{ only}\qquad\textbf{(B)}\ 11\text{ only}\qquad\textbf{(C)}\ 13\text{ only}\qquad\textbf{(D)}\ 101\qquad\textbf{(E)}\ 1001$

Solution

Problem 20

When simplified and expressed with negative exponents, the expression $(x+y)^{-1}(x^{-1}+y^{-1})$ is equal to:

$\textbf{(A)}\ x^{-2}+2x^{-1}y^{-1}+y^{-2}\qquad\textbf{(B)}\ x^{-2}+2^{-1}x^{-1}y^{-1}+y^{-2}\qquad\textbf{(C)}\ x^{-1}y^{-1}\qquad\textbf{(D)}\ \text{some even number}\qquad\textbf{(E)}\ \text{zero}$